\section{Introduction}

Drivers who want to minimize their trip time when traveling may drive
above the speed limit. This puts their lives, other drivers' lives and pedestrians' lives
in danger. In order to regulate unlawful speeding, police agencies deploy police units that
issue penalties to drivers that break the law. Our work models a game in which police agents
play a mixed-strategy approach in order to catch drivers that decide to go above the speed limit on a road.
The motivation to pursue this project is very clear and stems from the societal applications of the problem.
 We are interested in understanding how agents use decision theory and game theory to make decisions on
 which roads to speed and which not to speed. Additionally, we are interested in knowing what kinds of police deployment are most useful in the context of certain geographies.
 
 Our game is modelled as a graph. Driver agents seek to plan a path from goal to destination and traverse the
 graph to get to a destination. Police seek nodes that maximize their probability of finding speeding drivers.
 The game is modelled as a Bayesian Stackelberg game. The leader (the police agent) commits to a strategy first, and, given the police strategy, the follower(the driver agent) selfishly chooses, with a probability, the strategy that maximizes its profit. In turn, the leader may choose to play a follower Stackelberg strategy, in attempts of catching their follower. Our contribution is threefold: first we develop a graph-based model for the police and driver game. Second, we apply two learning algorithms to the model to evaluate the transient and dynamic interactions among the players. Third, we develop a full simulation that implements and tests the model and the learners.
 
 For the rest of this paper we discuss some background knowledge in section \ref{background} and we provide an account of related works in section \ref{litsurvey}. In section \ref{model} we present some of our design choices, and explain how we set up these choices and collected data in the \ref{implementation} section. We present results, and offer an analysis of these results in section \ref{results}. Ideas for future work are mentioned in section \ref{futurework} and concluding remarks in section \ref{conclusion}.
